### $(-1)$-enumeration of self-complementary plane partitions.

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The simple incidence structure $\mathcal{D}(\mathcal{A},2)$ formed by points and unordered pairs of distinct parallel lines of a finite affine plane $\mathcal{A}=(\mathcal{P},\mathcal{L})$ of order $n>2$ is a $2-({n}^{2},2n,2n-1)$ design. If $n=3$, $\mathcal{D}(\mathcal{A},2)$ is the complementary design of $\mathcal{A}$. If $n=4$, $\mathcal{D}(\mathcal{A},2)$ is isomorphic to the geometric design $A{G}_{3}(4,2)$ (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a $2-({n}^{2},2n,2n-1)$ design to be of the form $\mathcal{D}(\mathcal{A},2)$ for some finite affine plane $\mathcal{A}$ of order $n>4$. As a consequence we obtain a characterization of small designs $\mathcal{D}(\mathcal{A},2)$.

There is described a procedure which determines the quasigroup identity corresponding to a given 3-coloured 3-configuration with a simple edge basis.

We consider two classes of latin squares that are prolongations of Cayley tables of finite abelian groups. We will show that all squares in the first of these classes are confirmed bachelor squares, squares that have no orthogonal mate and contain at least one cell though which no transversal passes, while none of the squares in the second class can be included in any set of three mutually orthogonal latin squares.